Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

On the Rellich inequality with magnetic potentials

In lectures given in 1953 at New York University, Franz Rellich proved that for all f∈C₀^∞(Rⁿ \{0}) and n≠2where the constant C(n):=n²(n−4)²/16 is sharp. For n=2 extra conditions were required for f, and for n=4, C(4)=0, producing a trivial inequality. Influenced by recent work of Laptev-Weidl on Hardy-type inequalities in R², the authors show that for n≥2, the inclusion of a magnetic field B=curl(A) of Aharonov-Bohm type yields non-trivial Rellich-type inequalities of the formwhere Δ_A=(∇−iA)² is the magnetic Laplacian. As in the Laptev-Weidl inequality, the constant C(n,α) depends upon the distance of the magnetic flux to the integers Z. When the flux is an integer and α=0, the inequalities reduce to Rellich’s inequality.The first author gratefully acknowledges the hospitality and support of the Mathematics Department at UAB where much of this work was done.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

Finite domination, Novikov homology and nonsingular closed 1-forms

Let X be a finite connected CW-complex and ρ: a regular covering space with free abelian covering transformation group. For ξ ∈ H¹ (Xℝ) we define the notion of ξ-contractibility of X. This notion is closely related to the vanishing of the Novikov homology of the pair (X,ξ). We show that finite domination of is equivalent to X being ξ-contractible for every nonzero ξ with ρ*ξ =0  ∈ H¹(; ℝ). If M is a closed connected smooth manifold the condition that M is ξ-contractible is necessary for the existence of a nonsingular closed 1-form representing ξ. Also ξ-contractibility guarantees the definition of the Latour obstruction τ_L(M,ξ) whose vanishing is then sufficient for the existence of a nonsingular closed 1-form provided  dim M≥6. Now if ρ: is a finitely dominated regular ℤ^k-covering space we get that τ_L(M,ξ) is defined for every nonzero ξ with ρ*ξ=0 and the vanishing of one such obstruction implies the vanishing of all such τ_L(M,ξ).     

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λ_G is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of . Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST “Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the hospitality.     

Proof of a conjecture of José L. Rubio de Francia

Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1<p<∞, the characteristic function χ_I of an interval I in Γ is a p—multiplier with a uniform bound (independent of I) on the corresponding operator S_I on L^p(G). In this note it is shown that, for 1<p,q<∞, there is a constant C_p,q, independent of G and the particular ordering on Γ, such that for all sequences {I_j} of intervals in Γ and all sequences {f_j} in L^p(G). Such a result was conjectured by J.L. Rubio de Francia, who noted its validity when The present proof uses a transference argument, an approach which shows that any constant C_p,q for which the inequality holds when G = will serve for every G and every ordering on Γ. An added advantage of this approach is that it adapts to give an extension of the result for functions taking values in a UMD space.The work of the first author was partially supported by a grant from the National Science Foundation (U.S.A.). The second and third authors were partially supported by the HARP network HPRN-CT-2001-00273 of the European Commission and by grant BFM2001-0188 of Ministerio de Ciencia y Tecnologia.     

Holomorphic dynamics near germs of singular curves

Let M be a two dimensional complex manifold, p ∈ M and a germ of holomorphic foliation of M at p. Let be a germ of an irreducible, possibly singular, curve at p in M which is a separatrix for . We prove that if the Camacho-Sad-Suwa index Ind then there exists another separatrix for at p. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

Endomorphisms of H *(K(V, n); ) in the category of unstable modules

The ring of endomorphisms of the -cohomology of the Eilenberg-MacLane space K(V,n), in the category of unstable modules over the Steenrod algebra is calculated, where V is an elementary abelian 2-group, n is a non-negative integer and is the prime field of characteristic two. The result generalizes the theorem of Adams, Gunawardena and Miller, which corresponds to the case n=1.     

Power series rings and projectivity

We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A,) is any local ring, then A[[X]] behaves like a projective module in the sense that Ext^p_A(A[[X]], M)=0 for all -adically complete A-modules. The latter result is shown more generally for any flat A-module B instead of A[[X]]. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings. The authors were partly supported by NSERC grant 3-642-114-80 and by the DFG Schwerpunkt ``Global Methods in Complex Geometry'.     

Chow group of 1-cycles on the moduli space of vector bundles of rank 2 over a curve

Let X be a non-singular complex projective curve of genus ≥3. Choose a point x ∈ X. Let M_x be the moduli space of stable bundles of rank 2 with determinant We prove that the Chow group CH^Q₁(M_x) of 1-cycles on M_x with rational coefficients is isomorphic to CH^Q₀(X). By studying the rational curves on M_x, it is not difficult to see that there exits a natural homomorphism CH₀(J)→CH₁(M_x) where J denotes the Jacobian of X. The crucial point is to show that this homomorphism induces a homomorphism CH₀(X)→CH₁(M_x), namely, to go from the infinite dimensional object CH₀(J) to the finite dimensional object CH₀(X). This is proved by relating the degeneration of Hecke curves on M_x to the second term I^*2 of Bloch's filtration on CH₀(J). Insong Choe was supported by KOSEF (R01-2003-000-11634-0).     

Spherical harmonic analysis on affine buildings

Let be a locally finite regular affine building with root system R. There is a commutative algebra spanned by averaging operators A _λ , λ ∈ P ⁺, acting on the space of all functions f:V _P → , where V _P is in most cases the set of all special vertices of , and P ⁺ is a set of dominant coweights of R. This algebra is studied in [6] and [7] for à _n buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h: may be expressed in terms of the Macdonald spherical functions. We also provide a second formula for these homomorphisms in terms of an integral over the boundary of . We may regard as a subalgebra of the C ^*-algebra of bounded linear operators on ?²(V _P ), and we write for the closure of in this algebra. We study the Gelfand map , where M ₂= , and we compute M ₂ and the Plancherel measure of . We also compute the ?²-operator norms of the operators A _λ , λ ∈ P ⁺, in terms of the Macdonald spherical functions.     

Summing norms of identities between unitary ideals

Based on abstract interpolation, we prove asymptotic formulae for the (F,2)-summing norm of inclusions id: , where E and F are two Banach sequence spaces. Here, stands for the unitary ideal of operators on the n-dimensional Hilbert space whose singular values belong to E, and for the Hilbert-Schmidt operators. Our results are noncommutative analogues of results due to Bennett and Carl, as well as their recent generalizations to Banach sequence spaces. As an application, we give lower and upper estimates for certain s-numbers of the embeddings id: and id: . In the concluding section, we finally consider mixing norms. The second named author was supported by KBN Grant 2 P03A 042 18.     

Compactifications of contractible affine 3-folds into smooth Fano 3-folds with <Emphasis Type="Italic">B</Emphasis><Subscript>2</Subscript>=2

After the contributions of Furushima, Nakayama, Peternell and Schneider, in 1993, Furushima [Fur93] finally succeeded in the classification of the compactifications of the affine 3-space into smooth Fano 3-folds with B₂=1. In this paper, we consider the compactifications of the contractible affine 3-folds X (not necessarily X=) into smooth Fano 3-folds V with B₂=2. Consequently, we classify all such compactifications X↪(V,D₁∪D₂) in the case where K_V+D₁+D₂ is not nef. Furthermore, we see that infinitely many mutually non-isomorphic exotic 's can be compactified into Fano 3-folds with B₂=2. This phenomenon never occurs when B₂=1. During this research the author was supported as a Twenty-First Century COE Kyoto Mathematics Fellow.     

Mukai flops and deformations of symplectic resolutions

We prove that two projective symplectic resolutions of are connected by Mukai flops in codimension 2 for a finite sub-group G <Sp(2n). It is also shown that two projective symplectic resolutions of are deformation equivalent.     

Ribbon tableaux and the Heisenberg algebra

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: a Pieri and dual-Pieri formula for ribbon functions, a ribbon Murnaghan-Nakayama formula, ribbon Cauchy and dual Cauchy identities, and a -algebra isomorphism _n:(q)(q) which sends each to .Our study of the functions will be connected to the Fock space representation F of via a linear map :F(q) which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [KMS] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of . Identifying the Fock Space of H with the ring of symmetric functions (q) we will show that is in fact a map of H-modules with remarkable properties. The study of this map will lead to our identities concerning ribbon tableaux generating functions. We will also give a combinatorial proof that the ribbon Murnaghan-Nakayama and Pieri rules are formally equivalent.